A probability distribution P over X × {0, 1}. P can be defined in term of its marginal distribution over X , which we will denote by $P_X$ and the conditional labeling distribution, which is defined by the regression function

$$

µ(x) = P_{

(x,y)∼P}

(y = 1 | x)

$$

consider a 2-dimensional Euclidean domain, that is $X = R^2$, and the

following process of data generation: The marginal distribution over X is uniform over two square areas (1, 2) × (1, 2) ∪ (3, 4) × (1.5, 2.5). Points in the first

square Q1 = (1, 2) × (1, 2) are labeled 0 (blue) and points in the second square

Q2 = (3, 4) × (1.5, 2.5) are labeled 1 (red)

Describe the density function of $P_X$, and the regression function, Bayes predictor and Bayes risk of P.

In the image, I have defined the Probability density function. Is that correct? I want to find it over both the axis and in general in two dimensions.